Theoretical models and observations suggests that BH growth across cosmic times occurs mainly via gas accretion (e.g., [1]}).
A commonly adopted approach in cosmological models (SAMs and simulations) to describe the BH accretion process is to implement an approximated/sub-grid prescription, usually based on the Bondi–Hoyle–Lyttleton, spherically symmetric, approach [2]}, [3]}:
\(\dot{M}_{\rm BHL} = \alpha \frac{4 \pi G^2 M_{\rm BH}^2 \rho _{\rm gas}(r_B)}{(c_s^2 + v^2)^{3/2}}\) 
